It really depends on how much your very qualified renters want the house.
It might just appeal to them to have extra time to move their belongings. If they pay a portion of the rent and the utilities (which would be less because not everything will be in use for the entire month), offer the option of moving some of their belongings into the house during that month. It is an attractive offer because no one wants to move everything in one or two days It would give them time to do it at a slower, more productive pace. It helps them to be more organized. That scenario will appeal to the lady of the house, especially.
Put this offer to them in a positive light that will benefit them and you. I would just present it as a positive option. They may be able to save some moving costs if they can move things in a little at a time rather pay a moving company buy the hour or arrange family members to help over a weekend.
Let them know it is a win, win situation. See what happens. I would ask them to pay for 1/2 the rent and all of the utility bills required to move their things. Of course they must have the utilities transferred into their names. That will solidify the deal.
In the end it is all up to you. But if it were me, I would rather have the more qualified, desirable renters get off to a great start by helping them (and you) achieve the goal. WIN, WIN is always the best option.
Let them know a little about your situation and that you have another renter willing to pay it all.